Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces

TL;DR

This paper explores the connection between singularity categories of Brieskorn-Pham singularities and ACM bundles on Geigle-Lenzing spaces, introducing 2-extension bundles and constructing tilting objects with explicit algebraic properties.

Contribution

It introduces 2-extension bundles on Geigle-Lenzing spaces, establishes their relation to Cohen-Macaulay modules, and constructs a tilting object with a specific endomorphism algebra.

Findings

Established a correspondence between 2-extension bundles and Cohen-Macaulay modules.

Constructed a tilting object with endomorphism algebra as a 4-fold tensor product of Nakayama algebras.

Derived an explicit formula for the orbit number under Picard group action.

Abstract

We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating the stable category $\underline{\mathsf{ACM}} \, \mathbb{X}$ of arithmetically Cohen-Macaulay bundles on $\mathbb{X}$. We introduce the notion of $2$-extension bundles on $\mathbb{X}$, which is a higher dimensional analog of extension bundles on a weighted projective line of Geigle-Lenzing, and then establish a correspondence between $2$-extension bundles and a certain important class of Cohen-Macaulay $R$-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in $\underline{\mathsf{ACM}} \, \mathbb{X}$ consisting of $2$-extension bundles, whose endomorphism algebra is a $4$-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on $2$-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer.